If you want someone to think you're right, the best way to do is to make them think they came to that conclusion on their own. To that end, BBC Future suggests asking a few questions to lead them over to your side:

For example, "Why do you think this would be a good idea?" or "What do you think the advantages would be?" It sounds obvious, but letting someone persuade themselves will mean they are more confident of their decision in the long term – as if it had been their idea all along.

The flipside of this is you can watch out for people manipulating you when they ask these same types of questions. BBC Future covers a few other manipulation techniques to watch out for (or use, depending on your point of view), including using someone's field of view, the speed of your speech, and a subtle touch.